Quantum and Classical Disparity and Accord

TL;DR

This paper explores discrepancies and agreements between quantum and classical mechanics in simple systems, revealing non-locality, violations of classical principles, and introducing new concepts like partial well expectation values.

Contribution

It uncovers unexpected accords and violations in quantum-classical comparisons, and introduces the concept of expectation values over partial well widths.

Findings

Quantum and classical expectation values match for the SHO's second moment and beat period across all quantum numbers.

FPB energy states violate the Correspondence Principle at small quantum numbers.

Particle velocity in an infinite well varies regardless of box width or particle position.

Abstract

Discrepancies and accords between quantum (QM) and classical mechanics (CM) related to expectation values and periods are found for both the simple harmonic oscillator (SHO) and a free particle in a box (FPB), which may apply generally. These indicate non-locality is expected throughout QM. The FPB energy states violate the Correspondence Principle. Previously unexpected accords are found and proven that the classical and quantum expectation values are the same for the expectation value of the second moment and the beat period (i.e. beats between the phases for adjoining energy states) for the SHO for all quantum numbers, n. However, for the FPB the beat periods differ significantly at small n. It is shown that a particle's velocity in an infinite square well varies, no matter how wide the box, nor how far the particle is from the walls. The quantum free particle variances share an indirect commonality with the Aharonov-Bohm and Aharonov-Casher effects in that there is a quantum action in the absence of a force. The concept of an "Expectation Value over a Partial Well Width" is introduced. This paper raises the question as to whether these inconsistencies are undetectable, or can be empirically ascertained. These inherent variances may need to be fixed, or nature is manifestly more non-classical than expected.